The empty set is the set containing no elements.

In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. In axiomatic set theory it is postulated to exist by the axiom of empty set and all finite sets are constructed from it. The empty set is also sometimes called the null set, but because null set means something else in measure theory, that term is generally avoided in current work. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... This article or section is in need of attention from an expert on the subject. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ... In mathematics, a measure is a function that assigns a number, e. ...

Various possible properties of sets are trivially true for the empty set. In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ...

Notation

The empty set is denoted by either one of the symbols "" or "", derived from the letter Ø in the Danish and Norwegian alphabet, introduced by the Bourbaki group (specifically André Weil) in 1939. ^[1] Another common notation for the empty set is "{}". // The Ø (minuscule: ø), is a vowel and a letter used in the Danish, Faroese and Norwegian alphabets. ... The Danish and Norwegian alphabet is based upon the Latin alphabet and consists of 29 letters: (Listen to a Danish speaker recite the alphabet in Danish. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ...

Properties

(Here we use mathematical symbols.) The following table lists many specialized symbols commonly used in mathematics. ...

• For any set A, the empty set is a subset of A:

  ∀A: ∅ ⊆ A

• For any set A, the union of A with the empty set is A:

  ∀A: A ∪ ∅ = A

• For any set A, the intersection of A with the empty set is the empty set:

  ∀A: A ∩ ∅ = ∅

• For any set A, the Cartesian product of A and the empty set is empty:

  ∀A: A × ∅ = ∅

• The only subset of the empty set is the empty set itself:

  ∀A: A ⊆ ∅ ⇒ A = ∅

• The number of elements of the empty set (that is its cardinality) is zero; in particular, the empty set is finite:

  |∅| = 0

• For any property:
  • for every element of ∅ the property holds (vacuous truth)
  • there is no element of ∅ for which the property holds
• Conversely: if, for some property, the following two statements hold:
  • for every element of V the property holds
  • there is no element of V for which the property holds

then V = ∅

Mathematicians speak of "the empty set" rather than "an empty set". In set theory, two sets are equal if they have the same elements; therefore there can be only one set with no elements. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Vacuous truth is a special topic of first-order logic. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closed and open. All its boundary points (of which there are none) are in the empty set, and the set is therefore closed; while for every one of its points (of which there are again none), there is an open neighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the fact that every finite set is compact. In mathematics, the real line is simply the set of real numbers. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

The closure of the empty set is empty. This is known as "preservation of nullary unions." In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In mathematics, the arity of a function or an operator is the number of arguments or operands it takes, A function or operator can thus be described as unary, binary, ternary,etc. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Common problems

The empty set is not the same thing as nothing; it is a set with nothing inside it, and a set is something. This often causes difficulty among those who first encounter it. It may be helpful to think of a set as a bag containing its elements; an empty bag may be empty, but the bag itself certainly exists.

By the definition of subset, the empty set is a subset of any set A, as every element x of {} belongs to A. If it is not true that every element of {} is in A, there must be at least one element of {} that is not present in A. Since there are no elements of {} at all, there is no element of {} that is not in A, leading us to conclude that every element of {} is in A and that {} is a subset of A. Any statement that begins "for every element of {}" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set." A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... Vacuous truth is a special topic of first-order logic. ...

Axiomatic set theory

In the axiomatization of set theory known as Zermelo-Fraenkel set theory, the existence of the empty set is assured by the axiom of empty set. The uniqueness of the empty set follows from the axiom of extensionality. This article or section is in need of attention from an expert on the subject. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ...

Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = {x in A | x ≠ x}, which can be defined to be the empty set. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ...

Does it exist or is it necessary?

While the empty set is a standard and universally accepted concept in mathematics, some philosophers and logicians continue to debate its meaning and usefulness.

Jonathan Lowe has argued that while the idea "was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object." It is not clear that such an idea makes sense. "All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a set which has no members. We cannot conjure such an entity into existence by mere stipulation."^{[citation needed]} Jonathan Lowe (E.J. Lowe) (born 1950) is currently Professor of Philosophy and Chair of the Examination Board of the Department of Philosophy at the University of Durham, England. ...

In "To be is to be the value of a variable…", Journal of Philosophy, 1984 (reprinted in his book Logic, Logic and Logic), the late George Boolos has argued that we can go a long way just by quantifying plurally over individuals, without reifying sets as singular entities having other entities as members. ^{[citation needed]} George Stephen Boolos (September 4, 1940, New York City - May 27, 1996) was a philosopher and a mathematical logician. ... In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular values. ... Reification, also called hypostatization, is treating a concept, an abstraction, as if it were a real, concrete thing. ...

In a recent book Tom McKay has disparaged the "singularist" assumption that natural expressions using plurals can be analysed using plural surrogates, such as signs for sets. He argues for an anti-singularist theory which differs from set theory in that there is no analogue of the empty set, and there is just one relation, among, that is an analogue of both the membership and the subset relation.^{[citation needed]}

Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) can also be confusing. (Such operations are nullary operations.) For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). This may seem odd, since there are no elements of the empty set, so how could it matter whether they are added or multiplied (since “they” do not exist)? Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication. In mathematics, the arity of a function or an operator is the number of arguments or operands it takes, A function or operator can thus be described as unary, binary, ternary,etc. ... Addition is one of the basic operations of arithmetic. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... This article is about the number one. ... In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

Bounds

Since the empty set has no members, when it is considered as a subset of any ordered set, then any member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set.^[2] When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely negative infinity, denoted which is defined to be less than every other extended real number, and positive infinity, denoted which is defined to be greater than every other extended real number, then: ... In mathematics, the real line is simply the set of real numbers. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... In mathematics, the extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (pronounced plus infinity and minus infinity). These new elements are not real numbers. ... Positive Infinity are an alternative, metal, punk, emo band from Miami, Florida. ...

and

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for minimum and infimum. In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...

The empty set and zero

It was mentioned earlier that the empty set has zero elements, or that its cardinality is zero. The connection between the two concepts goes further however: in the standard set-theoretic definition of natural numbers, zero is defined as the empty set. 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... Several ways have been proposed to define the natural numbers using set theory. ...

Category theory

If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set is the unique initial object of the category of sets and functions. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, an empty or nullary function, is a function whose domain is the empty set. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

The empty set can be turned into a topological space in just one way (by defining the empty set to be open); this empty topological space is the unique initial object in the category of topological spaces with continuous maps. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

Use in linguistics

The empty set is also used in linguistics and particularly in language-teaching to denote a natural form (also colloquially named the dictionary form), which is generally the nominative singular for languages with declensions. It is used to emphasise that nothing should be added to the noun. However, this type of empty set is usually written with the same size as the other letters and so looks much more like a ø than like a ∅. Linguistics is the scientific study of language, which can be theoretical or applied. ... The nominative case is a grammatical case for a noun. ... In linguistics, grammatical number is a morphological category characterized by the expression of quantity through inflection or agreement. ... In linguistics, declension is the inflection of nouns, pronouns and adjectives to indicate such features as number (typically singular vs. ...

The empty set symbol is sometimes used in natural language syntax and morphology to represent morphemes that are not pronounced. For other uses, see Syntax (disambiguation). ... For other uses, see Morphology. ...

Set theory generally is a basic tool in formal semantics, so the empty set plays an important role in linguistics in this respect as well. In theoretical computer science formal semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation. ...

For denoting important spaces, see also, open box (␣) and verbatim in LaTeX. This does not cite any references or sources. ...

See also

• Inhabited set

In mathematics, a set A is inhabited if there exists an element This is distinct from the set being nonempty in many forms of intuitionistic logic. ...

References

1. ^ Earliest Uses of Symbols of Set Theory and Logic
2. ^ Elementary Real Analysis Thomson, Bruckner and Bruckner Page 9

• Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.